Stock market crashes like those in October 1987 and October 1997, the turbulent period around the Asian Crisis in 1998 through 1999, or the burst of the “dotcom bubble”, together with the severely volatile period after Sep. 11, 2001 constantly reminds financial engineers and risk managers how often extreme events actually happen in the real-world of financial markets. These observations have led to increased efforts to improve the flexibility and statistical reliability of existing models to capture the dynamics of economic variables. The history of probabilistic modeling of economic variables, and especially price processes, by means of stochastic processes goes back to Bachelier who suggested Brownian Motion as a candidate to describe the evolution of stock markets. Not until 70 years later that Black, F. and Scholes M. “The pricing of options and corporate liabilities”, The Journal of Political Economy, 81(3):637-654 (1973), and Merton, R. C., “The theory of rational option pricing”, Bell Journal of Economics and Management Sciences, 4(1):141-183 (1973), the disclosures of which are incorporated by reference, used Geometric Brownian Motion to describe the stock price movements in their famous solution of the option pricing problem. Their Nobel Prize winning work inspired the foundation of the arbitrage pricing theory based on the martingale approach described in Harrison, J. M. and Kreps, D. M. “Martingales and arbitrage in multi-period securities markets”, Journal of Economic Theory, 20:381-408 (1973), and subsequently by Harrison, J. M. and Pliska, S. R., “Martingales and stochastic integrals in the theory of continuous trading”, Stochastic Processes and their Applications, 11(3): 215-260 (1981), the disclosures of which are incorporated by reference.
The key observation, that pricing of derivatives has to be effected under a so called risk-neutral or equivalent martingale measure usually denoted as Q, which differs from the data generating “market measure”, usually denoted as P, has led to increasing research on what is called “implicit models”. Examples of implicit models include stochastic volatility models, local volatility models, martingale models for the short rate and implied volatility models. The common characteristic inherent to implicit models is that the model parameters are not determined through estimation from observations under the market measure, but through calibration on market prices of derivatives directly under the martingale measure. As a direct consequence, and at the same time, the main drawback of the calibration framework is that the market prices cannot be explained, they are just fitted. The prices of liquid market instruments are used for the calibration procedure and, consequently, are reproduced more or less perfectly. However, for exotic derivatives the prices derived from the implicit models differ substantially, since there is no market data on which to directly calibrate. Moreover, from an objective viewpoint, there is no way to determine which pricing model is the most reliable one given that the statistical fit to historical realization of the underlying is not taken into account.
An alternative approach to model price processes is pursued by econometricians. The goal of this approach is to provide the highest possible accuracy with respect to the empirical observations, or, in other words, to model the statistical characteristics of financial data. Thus, the focus of this approach lies in the statistical properties of historical realization and the quality of forecasts. However, in using these models, important aspects of derivative pricing are neglected.
Most econometric approaches neither present any risk neutral price processes nor are the markets defined in these models checked for the absence of arbitrage. Further, the econometric model approach assumes implicit or explicit knowledge of the statistical characteristics of financial data. Researches accept that financial return distributions are left-skewed and leptokurtic. This property of distributions was first studied in Fama, E., “The behavior of stock market prices”, Journal of Business, 38:34-105 (1965), and Mandelbrot, B. B., “New methods in statistical economics”, Journal of Political Economy, 71:421-440 (1963), and subsequently reported by various authors including in Rachev, S. T. and Mitnik, S., Stable Paretian Models in Finance, John Wiley & Sons (2000), the disclosures of which are incorporated by reference. A probability distribution is considered leptokurtic if the distribution exhibits Kurtosis, where the mass of the distribution is greater in the tails and is less in the center or body, when compared to a Normal distribution.
In addition, a probability distribution can be considered asymmetric if one side of the distribution is not a mirror image of the other side, when the distribution is divided at the maximum value point or the mean. Additionally, in time-series or longitudinal sections of return distributions, one observes volatility clustering, that is, calm periods, which are followed by highly volatile periods or vice versa. Autoregressive conditional heteroskedastic (ARCH) models and the extension to generalized ARCH (GARCH) models introduced in Engle, R., “Autoregressive conditional heteroskedasticity with estimates of the variance of united kingdom inflation”, Econometrica, 50:987-1007 (1982), and Bollerslev, T., “Generalized autoregressive conditional heteroskedasticity”, Journal of Econometrics, 31:307-327 (1986), the disclosures of which are incorporated by reference, have become standard tools in empirical finance. They are capable of capturing two important features that characterize time series of returns on financial assets: volatility clustering or conditional heteroskedasticity and excess leptokurtosis or heavy-tailedness. As mentioned above, both of these phenomena were already observed by Mandelbrot, but he focused on the second property and proposed the stable Paretian distribution as a model for the unconditional asset returns.
However, unconditional heavy-tails and GARCH phenomena are not unrelated. For example, Diebold, F. X., “Empirical Modeling of Exchange Rate Dynamics”, Springer (1998), the disclosure of which is incorporated by reference, demonstrates that the GARCH process driven by normally distributed innovations generate time series with heavy-tailed unconditional distributions, and in de Vries, C. G., “On the relation between GARCH and stable processes”, Journal of Econometrics, 48:313-324 (1991), and Groenendijk, P. A., Lucas, A., and de Vries, C. G., “A note on the relationship between GARCH and symmetric stable processes”, Journal of Empirical Finance, 2:253-264 (1995), the disclosures of which are incorporated by reference, show that certain GARCH processes can give rise to unconditional stable Paretian distributions.
When fitting GARCH models to return series, GARCH residuals still tend to be heavy tailed. To accommodate this, GARCH models with heavier conditional innovation distributions than those of the normal have been proposed, among them the Student's t-distribution and the generalized error distribution (GED). To allow for particularly heavy-tailed, conditional, and unconditional, return distributions, GARCH processes with non-normal stable Paretian error distributions have been considered. The class of stable Paretian distributions contains the normal distribution as a special case, but also allows for heavy-tailedness, i.e., infinite variance, and asymmetry. Neither the Student's nor the GED distribution share the latter property. The stable Paretian distribution also has the appealing property that the stable Paretian distributions is the only distribution that arises as a limiting distribution of sums of independently, identically distributed (iid) random variables. This is required when error terms are assumed to be the sum of all external effects that are not captured by the model.
An occasional objection against the use of the non-normal stable Paretian distribution is that the distribution has infinite variance. This seems to contradict empirical studies suggesting the existence of third or fourth moments for various financial return data. For this reason, an enhanced GARCH model with innovations which follow the smoothly truncated stable (STS) distribution have been introduced in Menn, C., Rachev, S. T., “Smoothly truncated stable distributions, GARCH models, and option pricing”, Mathematical Methods of Operations Research 69:411-438 (2009), the disclosure of which is incorporated by reference. While the STS distribution has finite values for all moments, the distribution has neither exact form of probability density function nor the characteristic function. Kim, Y. S., Rachev, S. T., Bianchi, M. L., Fabozzi, F., “Tempered stable and tempered infinitely divisible GARCH models”, Journal of Banking and Finance 34, 2096-2109 (2010), the disclosure of which is incorporated by reference, used the classes of tempered stable distributions which have the closed form of characteristic functions, for modeling the innovations. More precisely, as the innovation distribution, they used the CGMY distribution, as described in Carr, P., Geman, H., Madan, D., Yor, M., “The fine structure of asset returns: an empirical investigation”, Journal of Business 75, 305-332) (2002), the MTS distribution, as described in Kim, Y. S., Rachev, S. T., Chung, D. M., Bianchi, M. L., “The modified tempered stable distribution, GARCH-models and option pricing”, Probability and Mathematical Statistics 29:91-117) (2009), and the KR distribution , as described in Kim, Y. S., Rachev, S. T., Bianchi, M. L., and Fabozzi, F. J., “A new tempered stable distribution and its application to finance” (2008), as the innovation distribution, the disclosures of which are incorporated by reference. The main drawback of those tempered stable distributions is that the values of their exponential moments are finite only on some closed interval in the real line. In the parameter estimation and the Monte Carlo simulation, the innovation processes are bounded in this closed interval.